Exercise 7.C.8

Proof. Suppose ${\text{⟨⋅,⋅⟩}}_T$ is an inner product on $V$. Let $v\in \mathrm{null}<!--\; FUNCTION\; APPLICATION\; -->T$. Then

Therefore, by the definiteness property of inner products, $v=0$. Thus $\mathrm{null}<!--\; FUNCTION\; APPLICATION\; -->T=\{0\}$ and so $T$ is invertible. Suppose now that $v$ is an eigenvector of $T$ with eigenvalue $\lambda$. Then

Thus $\lambda \ge 0$. Hence all eigenvalues of $T$ are nonnegative. We but need to show that $T$ is self-adjoint and then $T$ will be positive by 7.35. We have

Hence $T$ is self-adjoint.

Conversely, suppose $T$ is an invertible positive operator. The positive-definiteness property of ${\text{⟨⋅,⋅⟩}}_T$ follows from the forward direction of the previous exercises. For additivity in the first slot, we have

Similarly, ${\text{⟨⋅,⋅⟩}}_T$ satisfies homogeneity in the first slot. For conjugate symmetry, we have

where the second line follows because $T$ is self-adjoint. Therefore ${\text{⟨⋅,⋅⟩}}_T$ is indeed an inner product on $V$. □